in terms of surface variables alone [1]}.
Moreover, the operator \(G(\eta )\)  is analytic with respect to \(\eta \)  for Lipschitz functions \(\eta \) 
and this provides an expansion of the Hamiltonian near the stationary solution \(w = (\eta ,\xi ) = 0\) 
which corresponds to a fluid at rest [2]}. The solution \(w=0\)  is an elliptic stationary point in dynamical systems terms.
In this Hamiltonian framework, perturbation calculations can be performed following general rules from Hamiltonian transformation theory,
including canonical transformations and reduction to normal forms, to derive asymptotic models for weakly nonlinear water waves
while preserving the Hamiltonian character of the original system (see [3]} for a review).
